Question

The projection of a vector on three coordinate axes are $6,-3,2$ respectively. The direction cosines of the vector are

• A. $(6,-3,2)$
• B. $(\frac{6}{5},\frac{-5}{5},\frac{2}{5})$
• C. $(\frac{6}{7},\frac{-3}{7},\frac{2}{7})$
• D. $(\frac{-6}{7},\frac{-3}{7},\frac{2}{7})$

Answer 1

The correct option is C $(\frac{6}{7},\frac{-3}{7},\frac{2}{7})$

If $\cos \alpha ,\cos \beta ,\cos \gamma$ are the direction cosines then $r\cos \alpha =6,r\cos \beta =-3,r\cos \gamma =2$
Squaring ,$r^2{\cos }^2\alpha =36,r^2{\cos }^2\beta =9,r^2{\cos }^2\gamma =4$
$\Rightarrow r^2{\cos }^2\alpha +r^2{\cos }^2\beta +r^2{\cos }^2\gamma =(36+9+4)=49$
$\Rightarrow r^2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )=49$
$\Rightarrow r^2=49,r=7$
$\therefore$ the direction cosines are $\frac{6}{7},\frac{-3}{7},\frac{2}{7}$